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Clay Mathematics Proceedings

Volume 10, 2008

A Short Survey of Cyclic Cohomology

Masoud Khalkhali

Dedicated with admiration and affection to Alain Connes

Abstract. This is a short survey of some aspects of Alain Connes’ contribu-tions to cyclic cohomology theory in the course of his work on noncommutativegeometry over the past 30 years.

Contents

1. Introduction 12. Cyclic cohomology 33. From K-homology to cyclic cohomology 104. Cyclic modules 135. The local index formula and beyond 166. Hopf cyclic cohomology 21References 28

1. Introduction

Cyclic cohomology was discovered by Alain Connes no later than 1981 and infact it was announced in that year in a conference in Oberwolfach [5]. I havereproduced the text of his abstract below. As it appears in his report, one ofConnes’ main motivations to introduce cyclic cohomology theory came from indextheory on foliated spaces. Let (V,F) be a compact foliated manifold and let V/Fdenote the space of leaves of (V,F). This space, with its natural quotient topology,is, in general, a highly singular space and in noncommutative geometry one usuallyreplaces the quotient space V/F with a noncommutative algebra A = C∗(V,F)called the foliation algebra of (V,F). It is the convolution algebra of the holonomygroupoid of the foliation and is a C∗-algebra. It has a dense subalgebra A =C∞(V,F) which plays the role of the algebra of smooth functions on V/F . Let Dbe a transversally elliptic operator on (V,F). The analytic index of D, index(D) ∈K0(A), is an element of theK-theory ofA. This should be compared with the family

2010 Mathematics Subject Classification. Primary 58B34; Secondary 19D55, 16T05, 18G30.

c© 2010 Masoud Khalkhali

1

2 MASOUD KHALKHALI

index theorem [1] where the analytic index of a family of fiberwise elliptic operatorsis an element of the K-theory of the base. Connes realized that to identify this classby a cohomological expression it would be necessary to have a noncommutativeanalogue of the Chern character, i.e., a map from K0(A) to a, then unknown,cohomology theory for the noncommutative algebra A. This theory, now knownas cyclic cohomology, would then play the role of the noncommutative analogueof de Rham homology of currents for smooth manifolds. Its dual version, cyclichomology, corresponds, in the commutative case, to de Rham cohomology.

Connes arrived at his definition of cyclic cohomology by a careful analysis of thealgebraic structures deeply hidden in the (super)traces of products of commutatorsof operators. These expressions are directly defined in terms of an elliptic operatorand its parametrix and give the index of the operator when paired with a K-theoryclass. In his own words [5]:

“The transverse elliptic theory for foliations requires as a preliminary step apurely algebraic work, of computing for a noncommutative algebra A the cohomologyof the following complex: n-cochains are multilinear functionsϕ(f0, . . . , fn) of f0, . . . , fn ∈ A where

ϕ(f1, . . . , f0) = (−1)nϕ(f0, . . . , fn)

and the boundary is

bϕ(f0, . . . , fn+1) = ϕ(f0f1, . . . , fn+1)− ϕ(f0, f1f2, . . . , fn+1) + · · ·+(−1)n+1ϕ(fn+1f0, . . . , fn).

The basic class associated to a transversally elliptic operator, for A = the algebraof the foliation, is given by:

ϕ(f0, . . . , fn) = Trace (εF [F, f0][F, f1] · · · [F, fn]), f i ∈ Awhere

F =

(0 QP 0

), ε =

(1 00 −1

),

and Q is a parametrix of P . An operation

S : Hn(A)→ Hn+2(A)is constructed as well as a pairing

K(A)×H(A)→ C

where K(A) is the algebraic K-theory of A. It gives the index of the operator fromits associated class ϕ. Moreover 〈e, ϕ〉 = 〈e, Sϕ〉, so that the important groupto determine is the inductive limit Hp = Lim

→H n(A) for the map S. Using the

tools of homological algebra the groups Hn(A,A∗) of Hochschild cohomology withcoefficients in the bimodule A∗ are easier to determine and the solution of theproblem is obtained in two steps:1) the construction of a map

B : Hn(A,A∗)→ Hn−1(A)and the proof of a long exact sequence

· · · → Hn(A,A∗)B→ Hn−1(A) S→ Hn+1(A) I→ Hn+1(A,A∗)→ · · ·

where I is the obvious map from the cohomology of the above complex to theHochschild cohomology;

A SHORT SURVEY OF CYCLIC COHOMOLOGY 3

2) the construction of a spectral sequence with E2 term given by the cohomologyof the degree −1 differential I B on the Hochschild groups Hn(A,A∗) and whichconverges strongly to a graded group associated to the inductive limit.

This purely algebraic theory is then used. For A = C∞(V ) one gets the de Rhamhomology of currents, and for the pseudo-torus, i.e. the algebra of the Kroneckerfoliation, one finds that the Hochschild cohomology depends on the Diophantine na-ture of the rotation number while the above theory gives H0

p of dimension 2 and H1p

of dimension 2, as expected, but from some remarkable cancellations.”

A full exposition of these results later appeared in two IHES preprints [6], andwere eventually published as [9]. With the appearance of [9] one could say thatthe first stage of the development of noncommutative geometry and specially cycliccohomology reached a stage of maturity. In the next few sections I shall try to givea quick and concise survey of some aspects of cyclic cohomology theory as theywere developed in [9]. The last two sections are devoted to developments in thesubject after [9] arising from the work of Connes.

It is a distinct honor and a great pleasure to dedicate this short survey of cycliccohomology theory as a small token of our friendship to Alain Connes, the originatorof the subject, on the occasion of his 60th birthday. It inevitably only covers partof what has been done by Alain in this very important corner of noncommutativegeometry. It is impossible to cover everything, and in particular I have left outmany important topics developed by him including, among others, the Godbillon-Vey invariant and type III factors [8], the transverse fundamental class for foliations[8], the Novikov conjecture for hyperbolic groups [18], entire cyclic cohomology [10],and multiplicative characteristic classes [12]. Finally I would like to thank FarzadFathi zadeh for carefully reading the text and for several useful comments, andArthur Greenspoon who kindly edited the whole text.

2. Cyclic cohomology

Cyclic cohomology can be defined in several ways, each shedding light on adifferent aspect of it. Its original definition [5, 9] was through a remarkable sub-complex of the Hochschild complex that we recall first. By algebra in this paperwe mean an associative algebra over C. For an algebra A let

Cn(A) = Hom(A⊗(n+1), C), n = 0, 1, . . . ,

denote the space of (n+1)-linear functionals on A. These are our n-cochains. TheHochschild differential b : Cn(A)→ Cn+1(A) is defined as

(bϕ)(a0, . . . , an+1) =

n∑

i=0

(−1)iϕ(a0, . . . , aiai+1, . . . , an+1)

+(−1)n+1ϕ(an+1a0, . . . , an).

The cohomology of the complex (C∗(A), b) is the Hochschild cohomology of A withcoefficients in the bimodule A∗.

The following definition is fundamental and marks the departure from Hochschildcohomology in [5, 9]:

Definition 2.1. An n-cochain ϕ ∈ Cn(A) is called cyclic if

ϕ(an, a0, . . . , an−1) = (−1)nϕ(a0, a1, . . . , an)

4 MASOUD KHALKHALI

for all a0, . . . , an in A. The space of cyclic n-cochains will be denoted by Cnλ (A).

Just why, of all possible symmetry conditions on cochains, the cyclic propertyis a reasonable choice is at first glance not at all clear.

Lemma 2.1. The space of cyclic cochains is invariant under the action of b, i.e.,b Cnλ (A) ⊂ Cn+1

λ (A) for all n ≥ 0.

To see this one introduces the operators λ : Cn(A)→ Cn(A) and b′ : Cn(A)→Cn+1(A) by

(λϕ)(a0, . . . , an) = (−1)nϕ(an, a0, . . . , an−1),

(b′ϕ)(a0, . . . , an+1) =n∑

i=0

(−1)iϕ(a0, . . . , aiai+1, . . . , an+1),

and checks that (1 − λ)b = b′(1 − λ). Since C∗λ(A) = Ker (1 − λ), the lemma is

proved.We therefore have a subcomplex of the Hochschild complex, called the cyclic

complex of A:

(1) C0λ(A)

b−→ C1λ(A)

b−→ C2λ(A)

b−→ · · · .

Definition 2.2. The cohomology of the cyclic complex (1) is the cyclic cohomologyof A and will be denoted by HCn(A), n = 0, 1, 2, . . . .

And that is Connes’ first definition of cyclic cohomology. A cocycle for thecyclic cohomology group HCn(A) is called a cyclic n-cocycle on A. It is an (n+1)-linear functional ϕ on A which satisfies the two conditions:

(1− λ)ϕ = 0, and bϕ = 0.

The inclusion of complexes

(2) (C∗λ(A), b) → (C∗(A), b)

induces a map I from cyclic cohomology to Hochschild cohomology:

I : HCn(A) −→ HHn(A), n = 0, 1, 2, . . . .

A closer inspection of the long exact sequence associated to (2), yields Connes’long exact sequence relating Hochschild cohomology to cyclic cohomology. This ishowever easier said than done. The reason is that to identify the cohomology ofthe quotient one must use another long exact sequence, and combine the two longexact sequences to obtain the result. To simplify the notation, let us denote theHochschild and cyclic complexes by C and Cλ, respectively. Then (2) gives us anexact sequence of complexes

(3) 0→ Cλ → Cπ→ C/Cλ → 0.

Its associated long exact sequence is

(4) · · · −→ HCn(A) −→ HHn(A) −→ Hn(C/Cλ) −→ HCn+1(A) −→ · · ·We need to identify the cohomology groups Hn(C/Cλ). To this end, consider theshort exact sequence of complexes

(5) 0 −→ C/Cλ1−λ−→ (C, b′)

N−→ Cλ −→ 0,

A SHORT SURVEY OF CYCLIC COHOMOLOGY 5

where the operator N is defined by

N = 1 + λ+ λ2 + · · ·+ λn : Cn −→ Cn.

The relations (1 − λ)b = b′(1 − λ), N(1 − λ) = (1 − λ)N = 0, and bN = Nb′

show that 1 − λ and N are morphisms of complexes in (5). As for the exactnessof (5), the only nontrivial part is to show that ker (N) ⊂ im (1 − λ), which can beverified. Assuming A is unital, the middle complex (C, b′) in (5) can be shown to beexact with a contracting homotopy s : Cn → Cn−1 defined by (sϕ)(a0, . . . , an−1) =(−1)n−1ϕ(a0, . . . , an−1, 1), which satisfies b′s + sb′ = id. The long exact sequenceassociated to (5) looks like(6)· · · −→ Hn(C/Cλ) −→ Hn

b′(C) −→ HCn(A) −→ Hn+1(C/Cλ) −→ Hn+1b′ (C) −→ · · ·

Since Hnb′(C) = 0 for all n, it follows that the connecting homomorphism

(7) δ : HCn−1(A)→ Hn(C/Cλ)

is an isomorphism for all n ≥ 0. Using this in (4), we obtain Connes’ long exactsequence relating Hochschild and cyclic cohomology:

(8) · · · −→ HCn(A) I−→ HHn(A) B−→ HCn−1(A) S−→ HCn+1(A) −→ · · · .

The operators B and S play a prominent role in noncommutative geometry.As we shall see, the operator B is the analogue of de Rham’s differential in thenoncommutative world, while the periodicity operator S is closely related to Bottperiodicity in topological K-theory. Remarkably, there is a formula for B on thelevel of cochains given by B = NB0, where B0 : Cn → Cn−1 is defined by

B0ϕ(a0, . . . , an−1) = ϕ(1, a0, . . . , an−1)− (−1)nϕ(a0, . . . , an−1, 1).

Using the relations (1− λ)b = b′(1− λ), (1− λ)N = N(1− λ) = 0, bN = Nb′, andsb′ + b′s = 1, one shows that

(9) bB +Bb = 0, and B2 = 0.

Using the periodicity operator S, the periodic cyclic cohomology of A is thendefined as the direct limit of cyclic cohomology groups under the operator S:

HP i(A) := Lim−→

HC2n+i(A), i = 0, 1.

Notice that since S has degree 2, there are only two periodic groups. These periodicgroups have better stability properties compared to cyclic cohomology groups. Forexample, they are homotopy invariant, and they pair with K-theory.

A much deeper relationship between Hochschild and cyclic cohomology groupsis encoded in Connes’ (b, B)-bicomplex and the associated Connes spectral sequencethat we shall briefly recall now. Consider the relations (9). The (b, B)-bicomplex

6 MASOUD KHALKHALI

of a unital algebra A, denoted by B (A), is the bicomplex

......

...

C2(A) B−−−−→ C1(A) B−−−−→ C0(A)

b

x b

x

C1(A) B−−−−→ C0(A)

b

x

C0(A)As usual, there are two spectral sequences attached to this bicomplex. The fol-

lowing fundamental result of Connes [9] shows that the spectral sequence obtainedfrom filtration by rows converges to cyclic cohomology. Notice that the E1 term ofthis spectral sequence is the Hochschild cohomology of A.

Theorem 2.1. The map ϕ 7→ (0, . . . , 0, ϕ) is a quasi-isomorphism of complexes

(C∗λ(A), b)→ (TotB (A), b+B).

This is a consequence of the vanishing of the E2 term of the second spectralsequence (filtration by columns) of B(A). To prove this, Connes considers the shortexact sequence of b-complexes

0 −→ ImB −→ KerB −→ KerB/ImB −→ 0,

and proves that ([9], Lemma 41), the induced map

Hb(ImB) −→ Hb(KerB)

is an isomorphism. This is a very technical result. It follows that Hb(KerB/ImB)vanishes. To take care of the first column one appeals to the fact that ImB ≃Ker (1 − λ) is the space of cyclic cochains.

We give an alternative proof of Theorem (2.1) above. To this end, consider thecyclic bicomplex C(A) defined by

......

...

C2(A) 1−λ−−−−→ C2(A) N−−−−→ C2(A) 1−λ−−−−→ · · ·xb

x−b′xb

C1(A) 1−λ−−−−→ C1(A) N−−−−→ C1(A) 1−λ−−−−→ · · ·xb

x−b′xb

C0(A) 1−λ−−−−→ C0(A) N−−−−→ C0(A) 1−λ−−−−→ · · ·The total cohomology of C(A) is isomorphic to cyclic cohomology:

Hn(Tot C (A)) ≃ HCn(A), n ≥ 0.

This is a consequence of the simple fact that the rows of C(A) are exact except indegree zero, where their cohomology coincides with the cyclic complex (C∗

λ(A), b).

A SHORT SURVEY OF CYCLIC COHOMOLOGY 7

So it suffices to show that TotB(A) and Tot C(A) are quasi-isomorphic. This canbe done by explicit formulas. Consider the chain maps

I : TotB(A) → TotC(A), I = id +Ns,

J : TotC(A) → TotB(A), J = id + sN.

It can be directly verified that the following operators define chain homotopy equiv-alences:

g : TotB(A) → TotB(A), g = Ns2B0,

h : Tot C(A) → Tot C(A), h = s.

To give an example of an application of the spectral sequence of Theorem(2.1), let me recall Connes’ computation of the continuous cyclic cohomology ofthe topological algebra A = C∞(M), i.e., the algebra of smooth complex valuedfunctions on a closed smooth n-dimensional manifoldM . This example is importantsince, apart from its applications, it clearly shows that cyclic cohomology is anoncommutative analogue of de Rham homology.

The continuous analogues of Hochschild and cyclic cohomology for topologicalalgebras are defined as follows [9]. Let A be a topological algebra. A continuouscochain on A is a jointly continuous multilinear map ϕ : A × A × · · · × A → C.By working with just continuous cochains, as opposed to all cochains, one obtainsthe continuous analogues of Hochschild and cyclic cohomology groups. In workingwith algebras of smooth functions (both in the commutative and noncommutativecase), it is essential to use this continuous analogue.

The topology of C∞(M) is defined by the sequence of seminorms

‖f‖n = sup |∂α f |, |α| ≤ n,

where the supremum is over a fixed, finite, coordinate cover for M . Under thistopology, C∞(M) is a locally convex, in fact nuclear, topological algebra. Similarlyone topologizes the space of p-forms on M for all p ≥ 0. Let

ΩpM := Homcont(ΩpM,C)

denote the continuous dual of the space of p−forms on M . Elements of ΩpM arecalled de Rham p-currents. By dualizing the de Rham differential d, we obtain adifferential d∗ : Ω∗M → Ω∗−1M , and a complex, called the de Rham complex ofcurrents on M :

Ω0Md∗←− Ω1M

d∗←− Ω2Md∗←− · · · .

The homology of this complex is the de Rham homology of M and we denote it byHdR

∗ (M).It is easy to check that for any m-current C, closed or not, the cochain

(10) ϕC(f0, f1, . . . , fm) := 〈C, f0df1 · · · dfm〉,is a continuous Hochschild cocycle on C∞(M). Now if C is closed, then one checksthat ϕC is a cyclic m-cocycle on C∞(M). Thus we obtain natural maps

(11) ΩmM → HHmcont(C

∞(M))

and

(12) ZmM → HCmcont(C∞(M)),

8 MASOUD KHALKHALI

where Zm(M) ⊂ ΩmM is the space of closed m-currents on M . For example, if Mis oriented and C represents its orientation class, then

ϕC(f0, f1, . . . , fn) =

∫

M

f0df1 · · · dfn,(13)

which is easily checked to be a cyclic n-cocycle on A.In [9], using an explicit resolution, Connes shows that (11) is a quasi-isomorph-

ism. Thus we have a natural isomorphism between space of de Rham currents onM and the continuous Hochschild cohomology of C∞(M) :

(14) HHicont(C

∞(M)) ≃ ΩiM i = 0, 1, . . .

To compute the continuous cyclic homology of A, one first observes that under theisomorphism (14) the operator B corresponds to the de Rham differential d∗. Moreprecisely, for each integer n ≥ 0 there is a commutative diagram:

Ωn+1Mµ−−−−→ Cn+1(A)

yd∗yB

ΩnMµ−−−−→ Cn(A)

where µ(C) = ϕC and ϕC is defined by (10). Then, using the spectral sequence ofTheorem (2.1) and the isomorphism (14), Connes obtains [9]:

(15) HCncont(C∞(M)) ≃ Zn(M)⊕HdR

n−2(M)⊕ · · · ⊕HdRk (M),

where k = 0 if n is even and k = 1 is n is odd. For the continuous periodic cycliccohomology he obtains

(16) HP kcont(C∞(M)) ≃

⊕

i

HdR2i+k(M), k = 0, 1.

We shall also briefly recall Connes’ computation of the Hochschild and cycliccohomology of smooth noncommutative tori [9]. This result already appeared inConnes’ Oberwolfach report [5]. When θ is rational, the smooth noncommutativetorus Aθ can be shown to be Morita equivalent to C∞(T 2), the algebra of smoothfunctions on the 2-torus. One can then use Morita invariance of Hochschild andcyclic cohomology to reduce the computation of these groups to those for the algebraC∞(T 2). This takes care of rational θ. So we can assume θ is irrational and wedenote the generators of Aθ by U and V with the relation V U = λUV , whereλ = e2πiθ.

Recall that an irrational number θ is said to satisfy a Diophantine condition if|1− λn|−1 = O(nk) for some positive integer k.

Proposition 2.1. ([9]) Let θ /∈ Q. Thena) One has HH0(Aθ) = C,b) If θ satisfies a Diophantine condition then HHi(Aθ) is 2-dimensional for i=1and is 1-dimensional for i = 2,c) If θ does not satisfy a Diophantine condition, then HHi(Aθ) are infinite dimen-sional non-Hausdorff spaces for i = 1, 2.

Remarkably, for all values of θ, the periodic cyclic cohomology is finite dimen-sional and is given by

HP 0(Aθ) = C2, HP 1(Aθ) = C2.

A SHORT SURVEY OF CYCLIC COHOMOLOGY 9

An explicit basis for these groups are given by cyclic 1-cocycles

ϕ1(a0, a1) = τ(a0δ1(a1)), and ϕ1(a0, a1) = τ(a0δ2(a1))

and cyclic 2-cocycles

ϕ(a0, a1, a2) = τ(a0(δ1(a1)δ2(a2)− δ2(a1)δ1(a2))), and Sτ,

where δ1, δ2 : Aθ → Aθ are the canonical derivations defined by

δ1(∑

amnUmV n) =

∑mamnU

mV n, δ2(UmV n) =

∑namnU

mV n,

and τ : Aθ → C is the canonical trace.A noncommutative generalization of formulas like (13) was introduced in [9]

and played an important role in the development of cyclic cohomology theory ingeneral. It gives a geometric meaning to the notion of a cyclic cocycle over analgebra and goes as follows. Let (Ω, d) be a differential graded algebra. A closedgraded trace of dimension n on (Ω, d) is a linear map

∫: Ωn −→ C

such that ∫dω = 0, and

∫[ω1, ω2] = 0,

for all ω in Ωn−1, ω1 in Ωi, ω2 in Ωj and i+ j = n. An n dimensional cycle over analgebra A is a triple (Ω,

∫, ρ), where

∫is an n-dimensional closed graded trace on

(Ω, d) and ρ : A→ Ω0 is an algebra homomorphism. Given a cycle (Ω,∫, ρ) over

A, its character is the cyclic n-cocycle on A defined by

(17) ϕ(a0, a1, . . . , an) =

∫ρ(a0)dρ(a1) · · · dρ(an).

Conversely one shows that all cyclic cocycles are obtained in this way.Once one has the definition of cyclic cohomology, it is not difficult to formulate

a dual notion of cyclic homology and a pairing between the two. Let Cn(A) =A⊗(n+1). The analogues of the operators b, b′ and λ are easily defined on C∗(A)and are usually denoted by the same letters, as we do here. For example b :Cn(A)→ Cn−1(A) is defined by

b(a0 ⊗ · · · ⊗ an) =

n−1∑

i=0

(−1)i(a0 ⊗ · · · ⊗ aiai+1 ⊗ · · · ⊗ an)(18)

+ (−1)n(ana0 ⊗ · · · ⊗ an−1).(19)

Let

Cλn(A) := Cn(A)/Im(1− λ).

The relation (1 − λ)b′ = b(1 − λ) shows that the operator b is well defined onCλ∗ (A). The complex (Cλ∗ (A), b) is called the homological cyclic complex of A andits homology, denoted by HCn(A), n = 0, 1, . . . , is the cyclic homology of A. Theevaluation map 〈ϕ, (a0 ⊗ · · · ⊗ an)〉 7→ ϕ(a0, . . . , an) clearly defines a degree zeropairingHC∗(A)⊗HC∗(A)→ C. Many results of cyclic cohomology theory, such asConnes’ long exact sequence and spectral sequence, and Morita invariance, continueto hold for cyclic homology theory with basically the same proofs.

Another important idea of Connes in the 1980’s was the introduction of entirecyclic cohomology of Banach algebras [10]. This allows one to deal with algebras of

10 MASOUD KHALKHALI

functions on infinite dimensional (noncommutative) spaces such as those appearingin constructive quantum field theory. These algebras typically don’t carry finitelysummable Fredholm modules, but in some cases have so-called θ-summable Fred-holm modules. In [10] Connes extends the definition of Chern character to suchFredholm modules with values in entire cyclic cohomology.

After the appearance of [9], cyclic (co)homology theory took on many livesand was further developed along distinct lines, including a purely algebraic one,with a big impact on algebraic K-theory. The cyclic cohomology of many algebraswas later computed including the very important case of group algebras [2] andgroupoid algebras. For many of these more algebraic aspects of the theory we referto [33, 32] and references therein.

3. From K-homology to cyclic cohomology

As I said in the introduction, Connes’ original motivation for the developmentof cyclic cohomology was to give a receptacle for a noncommutative Chern charactermap on the K-homology of noncommutative algebras. The cycles of K-homologycan be represented by, even or odd, Fredholm modules. Here we just focus on theodd case, and we refer to [9, 13] for the even case. Given a Hilbert space H, letL(H) denote the algebra of bounded linear operators on H, and K(H) denote thealgebra of compact operators. Also, for 1 ≤ p <∞, let Lp(H) denote the Schattenideal of p-summable operators. By definition, T ∈ Lp(H) if |T |p is a trace classoperator.

Definition 3.1. An odd Fredholm module over a unital algebra A is a pair (H, F )where1. H is a Hilbert space endowed with a representation

π : A −→ L(H),2. F ∈ L(H) is a bounded selfadjoint operator with F 2 = I,3. For all a ∈ A we have

(20) [F, π(a)] = Fπ(a)− π(a)F ∈ K(H).A Fredholm module (H, F ) is called p-summable if, instead of (20), we have

the stronger condition:

[F, π(a)] ∈ Lp(H)(21)

for all a ∈ A.To give a simple example, let A = C(S1) be the algebra of continuous functions

on the circle and let A act on H = L2(S1) as multiplication operators. Let F (en) =en if n ≥ 0 and F (en) = −en for n < 0, where en(x) = e2πinx, n ∈ Z, denotes thestandard orthonormal basis ofH. Clearly F is selfadjoint and F 2 = I. To show that[F, π(f)] is a compact operator for all f ∈ C(S1), notice that if f =

∑|n|≤N anen

is a finite trigonometric sum then [F, π(f)] is a finite rank operator and hence iscompact. In general we can uniformly approximate a continuous function by atrigonometric sum and show that the commutator is compact for any continuousf . This shows that (H, F ) is an odd Fredholm module over C(S1). This Fredholmmodule is not p-summable for any 1 ≤ p < ∞. If we restrict it to the subalgebraC∞(S1) of smooth functions, then it can be checked that (H, F ) is in fact p-summable for all p > 1, but is not 1-summable even in this case.

A SHORT SURVEY OF CYCLIC COHOMOLOGY 11

Now let me describe Connes’ noncommutative Chern character fromK-homologyto cyclic cohomology. Let (H, F ) be an odd p-summable Fredholm module over analgebra A. For any odd integer 2n − 1 such that 2n ≥ p, Connes defines a cyclic(2n− 1)-cocycle ϕ2n−1 on A by [9]

(22) ϕ2n−1(a0, a1, . . . , a2n−1) = Tr (F [F, a0][F, a1] · · · [F, a2n−1]),

where Tr denotes the operator trace and instead of π(a) we simply write a. Noticethat by our p-summability assumption, each commutator is in Lp(H) and hence,by Holder inequality for Schatten class operators, their product is in fact a traceclass operator as soon as 2n ≥ p. One checks by a direct computation that ϕ2n−1

is a cyclic cocycle.The next proposition shows that these cyclic cocycles are related to each other

via the periodicity S-operator of cyclic cohomology. This is probably how Connescame across the periodicity operator S in the first place.

Proposition 3.1. For all n with 2n ≥ p we have

Sϕ2n−1 = −(n+ 12 )ϕ2n+1.

By rescaling ϕ2n−1’s, one obtains a well defined element in the periodic cycliccohomology. The (unstable) odd Connes-Chern character Ch2n−1 = Ch2n−1(H, F )of an odd finitely summable Fredholm module (H, F ) over A is defined by rescalingthe cocycles ϕ2n−1 appropriately. Let

Ch2n−1 (a0, . . . , a2n−1) := (−1)n2(n− 12 ) · · · 12 Tr (F [F, a0][F, a1] · · · [F, a2n−1]).

Definition 3.2. The Connes-Chern character of an odd p-summable Fredholmmodule (H, F ) over an algebra A is the class of the cyclic cocycle Ch2n−1 in theodd periodic cyclic cohomology group HP odd(A).

By the above Proposition, the class of Ch2n−1 in HP odd(A) is independent ofthe choice of n.

Let us compute the character of the Fredholm module of the above Examplewith A = C∞(S1). By the above definition, Ch1 (H, F ) = [ϕ1] is the class of thefollowing cyclic 1-cocycle in HP odd(A) :

ϕ1 (f0, f1) = Tr (F [F, f0][F, f1]).

One can identify this cyclic cocycle with a local formula. We claim that

ϕ1 (f0, f1) =4

2πi

∫f0df1, for all f0, f1 ∈ A.

By linearity, It suffices to check this relation for basis elements f0 = em, f1 = enfor all m,n ∈ Z, which is easy to do.

The duality, that is, the bilinear pairing, between K-theory and K-homologyis defined through the Fredholm index. More precisely there is an index pairingbetween odd (resp. even) Fredholm modules over A and the algebraic K-theory

group Kalg1 (A) (resp. K0(A)). We shall describe it only in the odd case at hand.

Let (H, F ) be an odd Fredholm module over A and let U ∈ A× be an invertibleelement in A. Let P = F+1

2 : H → H be the projection operator defined by F .One checks that the operator

PUP : PH → PH

12 MASOUD KHALKHALI

is a Fredholm operator. In fact, using the compactness of commutators [F, a], onechecks that PU−1P is an inverse for PUP modulo compact operators, which ofcourse implies that PUP is a Fredholm operator. The index pairing is then definedas

〈(H, F ), [U ]〉 := index (PUP ),

where the index on the right hand side is the Fredholm index. If the invertibleU happens to be in Mn(A) we can apply this definition to the algebra Mn(A) bynoticing that (H⊗Cn, F ⊗ 1) is a Fredholm module over Mn(A) in a natural way.The resulting map can be shown to induce a well defined additive map

〈(H, F ), −〉 : Kalg1 (A)→ C.

Notice that this map is purely topological in the sense that to define it we did nothave to impose any finite summability, i.e., smoothness, condition on the Fredholmmodule.

Going back to our example and choosing f : S1 → GL1(C) a continuous func-

tion on S1 representing an element ofKalg1 (C(S1)), the index pairing 〈[(H, F )], [f ]〉 =

index(PfP ) can be explicitly calculated. In fact in this case a simple homotopyargument gives the index of the Toeplitz operator PfP : PH → PH in terms of thewinding number of f around the origin:

〈[(H, F )], [f ]〉 = −W (f, 0).

Of course, when f is smooth the winding number can be computed by integratingthe 1-form 1

2πidzz over the curve defined by f :

W (f, 0) =1

2πi

∫f−1df =

1

2πiϕ(f−1, f)

where ϕ is the cyclic 1-cocycle on C∞(S1) defined by ϕ(f, g) =∫fdg. This is

a special case of a very general index formula proved by Connes [9] in a fullynoncommutative situation:

Proposition 3.2. Let (H, F ) be an odd p-summable Fredholm module over analgebra A and let 2n− 1 be an odd integer such that 2n ≥ p. If u is an invertibleelement in A then

index (PuP ) =(−1)n22n

ϕ2n−1(u−1, u, . . . , u−1, u),

where the cyclic cocycle ϕ2n−1 is defined by

ϕ2n−1 (a0, a1, . . . , a2n−1) = Tr (F [F, a0][F, a1] · · · [F, a2n−1]).

The above index formula can be expressed in a more conceptual manner onceConnes’ Chern character in K-theory is introduced. In [4, 9], Connes shows thatthe Chern-Weil definition of Chern character on topological K-theory admits a vastgeneralization to a noncommutative setting. For a noncommutative algebra A andeach integer n ≥ 0, he defined pairings between cyclic cohomology and K-theory:

(23) HC2n(A) ⊗K0(A) −→ C, HC2n+1(A) ⊗Kalg1 (A) −→ C.

These pairings are compatible with the periodicity operator S in cyclic cohomologyin the sense that

〈[ϕ], [e]〉 = 〈S[ϕ], [e]〉,

A SHORT SURVEY OF CYCLIC COHOMOLOGY 13

for all cyclic cocycles ϕ and K-theory classes [e], and thus induce a pairing

HP i(A) ⊗Kalgi (A) −→ C, i = 0, 1

between periodic cyclic cohomology and K-theory.We briefly recall its definition. Let ϕ be a cyclic 2n-cocycle on A and let

e ∈Mk(A) be an idempotent representing a class inK0(A). The pairingHC2n(A)⊗K0(A) −→ C is defined by

(24) 〈[ϕ], [e]〉 = (n!)−1 ϕ(e, . . . , e),

where ϕ is the ‘extension’ of ϕ to Mk(A) defined by the formula

(25) ϕ(m0 ⊗ a0, . . . ,m2n ⊗ a2n) = tr(m0 · · ·m2n)ϕ(a0, . . . , a2n).

It can be shown that ϕ is a cyclic n-cocycle as well.The formulas in the odd case are as follows. Given an invertible matrix u ∈

Mk(A), representing a class in Kalg1 (A), and an odd cyclic (2n − 1)-cocycle ϕ on

A, the pairing is given by

(26) 〈[ϕ], [u]〉 := 2−(2n+1)

(n− 12 ) · · · 12

ϕ(u−1 − 1, u− 1, . . . , u−1 − 1, u− 1).

Any cyclic cocycle can be represented by a normalized cocycle for which ϕ(a0, . . . , an) =0 if ai = 1 for some i. When ϕ is normalized, formula (26) reduces to a particularlysimple form:

(27) 〈[ϕ], [u]〉 = 2−(2n+1)

(n− 12 ) · · · 12

ϕ(u−1, u, . . . , u−1, u).

Using the pairingHC2n−1(A)⊗Kalg1 (A)→ C and the definition of Ch2n−1 (H, F ),

the above index formula in Proposition (3.2) can be written as

(28) index (PuP ) = 〈Ch2n−1 (H, F ), [u]〉,or in its stable form

index (PuP ) = 〈Chodd (H, F ), [u]〉.This equality amounts to the equality Topological Index = Analytic Index in a fullynoncommutative setting.

An immediate consequence of the index formula (28) is an integrality theoremfor numbers defined by the right hand side of (28). This should be comparedwith classical integrality results for topological invariants of manifolds that areestablished through the Atiyah-Singer index theorem. An early nice applicationwas Connes’ proof of the idempotent conjecture for group C∗-algebras of free groupsin [9]. Among other applications I should mention a mathematical treatment ofintegral quantum Hall effect, and most recently to quantum computing in the workof Mike Freedman and collaborators [29].

4. Cyclic modules

With the introduction of the cyclic category Λ in [7], Connes took another majorstep in conceptualizing and generalizing cyclic cohomology far beyond its originalinception. We already saw in the last section three different definitions of the cycliccohomology of an algebra through explicit complexes. The original motivation of[7] was to define the cyclic cohomology of algebras as a derived functor. Sincethe category of algebras and algebra homomorphisms is not an additive category,

14 MASOUD KHALKHALI

the standard abelian homological algebra is not applicable here. Let k be a unitalcommutative ring. In [7], an abelian category Λk of cyclic k-modules is definedthat can be thought of as the ‘abelianization’ of the category of k-algebras. Cycliccohomology is then shown to be the derived functor of the functor of traces, as weshall explain in this section. More generally Connes defined the notion of a cyclicobject in an abelian category and its cyclic cohomology [7].

Later developments proved that this extension of cyclic cohomology was of greatsignificance. Apart from earlier applications, we should mention the recent work[16] where the abelian category of cyclic modules plays a role similar to that of thecategory of motives for noncommutative geometry. Another recent example is thecyclic cohomology of Hopf algebras [20, 21, 30, 31], which cannot be defined as thecyclic cohomology of an algebra or a coalgebra but only as the cyclic cohomologyof a cyclic module naturally attached to the given Hopf algebra and a coefficientsystem (see the last section for more on Hopf cyclic cohomology). Let us brieflysketch the definition of the cyclic category Λ.

Recall that the simplicial category ∆ is a small category whose objects are thetotally ordered sets

[n] = 0 < 1 < · · · < n, n = 0, 1, 2, . . . ,

and whose morphisms f : [n] → [m] are order preserving, i.e. monotone non-decreasing, maps f : 0, 1, . . . , n → 0, 1, . . . ,m. Of particular interest amongthe morphisms of ∆ are faces δi and degeneracies σj ,

δi : [n− 1]→ [n], i = 0, 1, . . . , n,

σj : [n+ 1]→ [n], j = 0, 1, . . . , n.

By definition δi is the unique injective morphism missing i and σj is the uniquesurjective morphism identifying j with j + 1.

The cyclic category Λ has the same set of objects as ∆ and in fact contains∆ as a subcategory. Morphisms of Λ are generated by simplicial morphisms andnew morphisms τn : [n] → [n], n ≥ 0, defined by τn(i) = i + 1 for 0 ≤ i < n andτn(n) = 0. We have the following extra relations:

τnδi = δi−1τn−1, τnδ0 = δn, 1 ≤ i ≤ n,

τnσi = σi−1τn+1, τnσ0 = σnτ2n+1 1 ≤ i ≤ n,

τn+1n = id.

It can be shown that the classifying space BΛ of the small category Λ is homotopyequivalent to the classifying space of the circle S1 [7].

A cyclic object in a category C is a functor Λop → C. A cocyclic object in C isa functor Λ → C. For any commutative unital ring k, we denote the category ofcyclic k-modules by Λk. A morphism of cyclic k-modules is a natural transformationbetween the corresponding functors. It is clear that Λk is an abelian category. Moregenerally, if A is an abelian category then the category ΛA of cyclic objects in Ais itself an abelian category.

Let Algk denote the category of unital k-algebras and unital algebra homomor-phisms. There is a functor

: Algk −→ Λk, A 7→ A,

defined by

An = A⊗(n+1), n ≥ 0,

A SHORT SURVEY OF CYCLIC COHOMOLOGY 15

with face, degeneracy and cyclic operators given by

δi(a0 ⊗ a1 ⊗ · · · ⊗ an) = a0 ⊗ · · · ⊗ aiai+1 ⊗ · · · ⊗ an,

δn(a0 ⊗ a1 ⊗ · · · ⊗ an) = ana0 ⊗ a1 ⊗ · · · ⊗ an−1,

σi(a0 ⊗ a1 ⊗ · · · ⊗ an) = a0 ⊗ · · · ⊗ ai ⊗ 1⊗ · · · ⊗ an,

τn(a0 ⊗ a1 ⊗ · · · ⊗ an) = an ⊗ a0 ⊗ · · · ⊗ an−1.

A unital algebra map f : A→ B induces a morphism of cyclic modules f : A → B

by f (a0 ⊗ · · · ⊗ an) = (f(a0)⊗ · · · ⊗ f(an)).This functor embeds the non-additive category of k-algebras into the abelian

category of cyclic k-modules. A first main observation of [7] is that

HomΛk(A, k) ≃ T (A),

where T (A) is the space of traces from A→ k. To a trace τ one associate the cyclicmap (fn)n≥0, where

fn(a0 ⊗ a1 ⊗ · · · ⊗ an) = τ(a0a1 · · · an), n ≥ 0.

It can be easily shown that this defines a one to one correspondence.Now we can state the following fundamental theorem of Connes [7] which

greatly extends the above observation and shows that cyclic cohomology is a de-rived functor, in fact an Ext functor, provided that we work in the category ofcyclic modules:

Theorem 4.1. Let k be a field of characteristic zero. For any unital k-algebra A,there is a canonical isomorphism

HCn(A) ≃ ExtnΛk(A, k), for all n ≥ 0.

Apart from their applications in the study of cyclic cohomology of algebras andHopf algebras (about the latter see the next section), cyclic modules have also cometo play an important role in applications of noncommutative geometry to numbertheory. They play a role similar to that of motives in algebraic geometry. Let mebriefly explain this point.

The program outlined by Connes, Consani and Marcolli in [16] aims at creatingan environment where something like Weil’s proof of the Riemann hypothesis forfunction fields can be repeated in the characteristic zero case. Among other things,they produce an analogue of the Frobenius automorphism in characteristic zero inthis paper. Since Connes’ trace formula is over the noncommutative adeles classspace [14], the geometric setting is that of noncommutative geometry and they mustgo far beyond what is done so far in noncommutative geometry and import manyideas from modern algebraic geometry to noncommutative geometry. To achievethis, as a first step, good analogues of etale cohomology, the category of motives,and correspondences in noncommutative geometry must be introduced. Happily itturns out that Connes’ category of cyclic modules and the closely related bivariantcyclic homology, as well as KK-theory, are quite useful in this regard.

The construction of the Frobenius in characteristic zero follows a very generalprocess that combines cyclic homology with quantum statistical mechanics in anovel way. Starting from a pair (A,ϕ) of an algebra and a state ϕ (a noncommu-tative space endowed with a ‘probability measure’), they proceed by invoking thecanonical one-parameter group of automorphisms σ = σϕ and consider the extremal

16 MASOUD KHALKHALI

equilibrium states Σβ at inverse temperatures β > 1. Under suitable conditions,there is an algebra map

ρ : A⋊σ R→ S(Σβ × R∗+)⊗ L,

where L denotes the algebra of trace class operators. The cyclic module D(A,ϕ) isdefined as the cokernel of the induced map by Tr ρ on the cyclic modules of thesetwo algebras. The dual multiplicative group R∗

+ acts on D(A,ϕ) and, in examplescoming from number theory, replaces Frobenius in characteristic zero. The threesteps involved in the construction of D(A,ϕ) are called cooling, distillation, anddual action in the paper.

A remarkable property of the cyclic category Λ, not shared by the simplicialcategory, is its self-duality in the sense that there is a natural isomorphism ofcategories Λ ≃ Λop [7]. Roughly speaking, the duality functor Λop −→ Λ acts as theidentity on objects of Λ and exchanges face and degeneracy operators while sendingthe cyclic operator to its inverse. Thus to a cyclic (resp. cocyclic) module one canassociate a cocyclic (resp. cyclic) module by applying the duality isomorphism.This duality plays an important role in Hopf cyclic cohomology.

5. The local index formula and beyond

In practice, computing Connes-Chern characters defined by formulas like (22)is rather difficult since they involve the ordinary operator trace and are non-local.Thus one needs to compute the class of this cyclic cocycle by a local formula. This israther similar to passing from the McKean-Singer formula for the index of an ellip-tic operator to a local cohomological formula involving integrating a locally defineddifferential form, i.e., the Atiyah-Singer index formula. The solution of this problemwas arrived at in two stages. First, in [13], Connes gave a partial answer by givinga local formula for the Hochschild class of the Chern character, and then Connesand Moscovici gave a formula that captures the full cyclic cohomology class of thecharacter by a local formula [19]. Broadly speaking, the ideas involved amount togoing from noncommutative differential topology to noncommutative spectral geom-etry, and need the introduction of two new concepts.

In the first place, a noncommutative analogue of integration was found byConnes by replacing the operator trace by the Dixmier trace [11], and, secondly,one refines the topological notion of Fredholm module by the metric notion of aspectral triple, or K-cycles as they were originally named in [13]. Developing thenecessary tools to handle this local index formula, shaped, more or less, the secondstage of the development of noncommutative geometry after the appearance of thelandmark papers [9]. One can say that while in its first stage noncommutativegeometry was influenced by differential and algebraic topology, especially indextheory, the Novikov conjecture and the Baum-Connes conjecture, in this secondstage it was chiefly informed by spectral geometry.

We start with a quick review of the Dixmier trace and the noncommutativeintegral, following [13] closely. For a compact operator T , let µn(T ), n = 1, 2, . . . ,

denote the sequence of eigenvalues of |T | = (T ∗T )12 written in decreasing order.

Thus, by the minimax principle, µ1(T ) = ||T ||, and in general

µn(T ) = inf ||T |V ||, n ≥ 1,

where the infimum is over the set of subspaces of codimension n − 1, and T |Vdenotes the restriction of T to the subspace V . The natural domain of the Dixmier

A SHORT SURVEY OF CYCLIC COHOMOLOGY 17

trace is the set of operators

L1,∞(H) := T ∈ K(H);N∑

1

µn(T ) = O (logN).

Notice that trace class operators are automatically in L1,∞(H). The Dixmier traceof an operator T ∈ L1,∞(H) measures the logarithmic divergence of its ordinarytrace. More precisely, we are interested in taking some kind of limit of the boundedsequence

σN (T ) =

∑N1 µn(T )

logNas N →∞. The problem of course is that, while by our assumption the sequence isbounded, the usual limit may not exists and must be replaced by a carefully chosen‘generalized limit’.

To this end, let TraceΛ(T ),Λ ∈ [1,∞), be the piecewise affine interpolation of

the partial trace function TraceN (T ) =∑N

1 µn(T ). Recall that a state on a C∗-algebra is a non-zero positive linear functional on the algebra. Let ω : Cb[e,∞)→ C

be a normalized state on the algebra of bounded continuous functions on [e,∞) suchthat ω(f) = 0 for all f vanishing at∞. Now, using ω, the Dixmier trace of a positiveoperator T ∈ L1,∞(H) is defined as

Trω(T ) := ω(τΛ(T )),

where

τΛ(T ) =1

logΛ

∫ Λ

e

Tracer(T )

log r

dr

r

is the Cesaro mean of the function Tracer(T )logr over the multiplicative group R∗

+. One

then extends Trω to all of L1,∞(H) by linearity.The resulting linear functional Trω is a positive trace on L(1,∞)(H). It is easy

to see from its definition that if T actually happens to be a trace class operatorthen Trω(T ) = 0 for all ω, i.e., the Dixmier trace is invariant under perturbationsby trace class operators. This is a very useful property and makes Trω a flexibletool in computations. The Dixmier trace, Trω, in general depends on the limitingprocedure ω; however, for the class of operators T for which LimΛ→∞ τΛ(T ) exit,it is independent of the choice of ω and is equal to LimΛ→∞τΛ(T ). One of themain results proved in [11] is that if M is a closed n-dimensional manifold, E is asmooth vector bundle on M , P is a pseudodifferential operator of order −n actingbetween L2-sections of E, and H = L2(M,E), then P ∈ L(1,∞)(H) and, for anychoice of ω, Trω(P ) = n−1Res(P ). Here Res denotes Wodzicki’s noncommutativeresidue. For example, if D is an elliptic first order differential operator, |D|−n isa pseudodifferential operator of order −n and, for any bounded operator a, theDixmier trace Trω(a|D|−n) is independent of the choice of ω.

The second ingredient of the local index formula is the notion of spectral triple[13]. Spectral triples provide a refinement of Fredholm modules. Going from Fred-holm modules to spectral triples is similar to going from the conformal class of aRiemannian metric to the metric itself. Spectral triples simultaneously provide anotion of Dirac operator in noncommutative geometry, as well as a Riemanniantype distance function for noncommutative spaces.

To motivate the definition of a spectral triple, we recall that the Dirac opera-tor D/ on a compact Riemannian Spinc manifold acts as an unbounded selfadjoint

18 MASOUD KHALKHALI

operator on the Hilbert space L2(M,S) of L2-spinors on the manifold M . If we letC∞(M) act on L2(M,S) by multiplication operators, then one can check that forany smooth function f , the commutator [D, f ] = Df − fD extends to a boundedoperator on L2(M,S). Now the geodesic distance d on M can be recovered fromthe following beautiful distance formula of Connes [13]:

d(p, q) = Sup|f(p)− f(q)|; ‖ [D, f ] ‖≤ 1, ∀p, q ∈M.

The triple (C∞(M), L2(M,S), D/) is a commutative example of a spectral triple. Itsgeneral definition, in the odd case, is as follows. This definition should be comparedwith Definition (3.1).

Definition 5.1. Let A be a unital algebra. An odd spectral triple on A is a triple(A,H, D) consisting of a Hilbert space H, a selfadjoint unbounded operator D :Dom(D) ⊂ H → H with compact resolvent, i.e., (D + λ)−1 ∈ K(H), for all λ /∈ R,and a representation π : A → L(H) of A such that for all a ∈ A, the commutator[D, π(a)] is defined on Dom(D) and extends to a bounded operator on H.

The finite summability assumption (21) for Fredholm modules has a finer ana-logue for spectral triples. For simplicity we shall assume that D is invertible (ingeneral, since KerD is finite dimensional, by restricting to its orthogonal comple-ment we can always reduce to this case). A spectral triple is called finitely summableif for some n ≥ 1

(29) |D|−n ∈ L1,∞(H).A simple example of an odd spectral triple is (C∞(S1), L2(S1), D), where D

is the unique selfadjoint extension of the operator −i ddx . Eigenvalues of |D| are|n|, n ∈ Z, which shows that, if we restrict D to the orthogonal complement ofconstant functions, then |D|−1 ∈ L1,∞(L2(S1)).

Given a spectral triple (A,H, D), one obtains a Fredholm module (A,H, F ) bychoosing F = Sign (D) = D|D|−1. Connes’ Hochschild character formula gives alocal expression for the Hochschild class of the Connes-Chern character of (A,H, F )in terms of D itself. For this one has to assume that the spectral triple (A,H, D)is regular in the sense that for all a ∈ A,

a and [D, a] ∈ ∩Dom(δk)

where the derivation δ is given by δ(x) = [|D|, x].Now, assuming (29) holds, Connes defines an (n+1)−linear functional ϕ on A

by

ϕ(a0, a1, . . . , an) = Trω(a0[D, a1] · · · [D, an]|D|−n).

It can be shown that ϕ is a Hochschild n-cocycle on A. We recall that a Hochschildn-cycle c ∈ Zn(A,A) is an element c =

∑a0 ⊗ a1 ⊗ · · · ⊗ an ∈ A⊗(n+1) such that

its Hochschild boundary b(c) = 0, where b is defined by (18). The following result,known as Connes’ Hochschild character formula, computes the Hochschild class ofthe Chern charcater by a local formula, i.e., in terms of ϕ:

Theorem 5.1. Let (A,H, D) be a regular spectral triple. Let F = Sign (D) denotethe sign of D and τn ∈ HCn(A) denote the Connes-Chern charcater of (H, F ). Forevery n-dimensional Hochschild cycle c =

∑a0⊗a1⊗· · ·⊗an ∈ Zn(A,A), one has

〈τn, c〉 =∑

ϕ(a0, a1, . . . , an).

A SHORT SURVEY OF CYCLIC COHOMOLOGY 19

Identifying the full cyclic cohomology class of the Connes-Chern character of(A,H, D) by a local formula is the content of Connes-Moscovici’s local index for-mula. For this we have to assume the spectral triple satisfies another technicalcondition. Let B denote the subalgebra of L(H) generated by operators δk(a) andδk([D, a]), k ≥ 1. A spectral triple is said to have a discrete dimension spectrum Σif Σ ⊂ C is discrete and for any b ∈ B the function

ζb(z) = Trace(b|D|−z), Re z > n,

extends to a holomorphic function on C \Σ. It is further assumed that Σ is simplein the sense that ζb(z) has only simple poles in Σ.

The local index formula of Connes and Moscovici [19] is given by the followingTheorem (we have used the formulation in [15]):

Theorem 5.2. 1. The equality∫−P = Resz=0 Trace(P |D|−z)

defines a trace on the algebra generated by A, [D,A], and |D|z , z ∈ C.2. There are only a finite number of non-zero terms in the following formula whichdefines the odd components (ϕn)n=1,3,... of an odd cyclic cocycle in the (b, B) bi-complex of A: For each odd integer n let

ϕn(a0, . . . , an) :=

∑

k

cn,k

∫−a0[D, a1](k1) · · · [D, an](kn)|D|−n−2|k|

where T (k) := ∇k and ∇(T ) = D2T − TD2, k is a multi-index, |k| = k1 + · · ·+ knand

cn,k := (−1)|k|√2i(k1! · · · kn!)−1((k1 + 1) · · · (k1 + k2 + · · · kn))−1Γ(|k|+ n

2).

3. The pairing of the cyclic cohomology class (ϕn) ∈ HC∗(A) with K1(A) gives theFredholm index of D with coefficients in K1(A).

As is indicated in part 1) of the above Theorem, a regular spectral triple nec-essarily defines a trace on its underlying algebra by the formula a ∈ A 7→

∫−a =

Resz=0 Trace(a|D|−z). Thus, to deal with ‘type III algebras’ which carry no non-trivial traces, the notion of spectral triple must be modified. In [25] Connes andMoscovici define a notion of twisted spectral triple, where the twist is afforded byan algebra automorphism (related to the modular automorphism group). Moreprecisely, one postulates that there exists an automorphism σ of A such that thetwisted commutators

[D, a]σ := Da− σ(a)D

are bounded operators for all a ∈ A. They show that, in the twisted case, theDixmier trace induces a twisted trace on the algebra A, but surprisingly, undersome regularity conditions, the Connes-Chern character of the phase space landsin ordinary cyclic cohomology. Thus its pairing with ordinary K-theory makessense, and it can be recovered as the index of Fredholm operators. This suggeststhe significance of developing a local index formula for twisted spectral triples, i.e.finding a formula for a cocycle, cohomologous to the Connes-Chern character inthe (b, B)-bicomplex, which is given in terms of twisted commutators and residuefunctionals. I beleive that this new theme of twisted spectral triples, and type IIInoncommutative geometry in general, will dominate the subject in near future.

20 MASOUD KHALKHALI

For example, very recently a local index formula has been proved for a class oftwisted spectral triples by Henri Moscovici [34] that can be found in the presentvolume. This class is obtained by twisting an ordinary spectral triple (A,H, D)by a subgroup G of conformal similarities of the triple, i.e. the set of all unitaryoperators U ∈ U(H) such that UAU∗ = A, and UDU∗ = µ(U)D, with µ(U) > 0.It is shown that the crossed product algebra A ⋊ G admits an automorphism σ,given by the formula σ(aU) = µ(U)−1aU , for all a ∈ A, U ∈ G, and (A⋊G,H, D)is a twisted spectral triple. The analogue of the noncommutative residue on thecircle, for algebras of formal twisted pseudodifferential symbols, is constructed in[27].

A very recent development related to (twisted) spectral triples is the noncom-mutative Gauss-Bonnet theorem of Connes and Tretkoff for the noncommutativetwo-torus Aθ [26]. In classical geometry a spectral zeta function is associated to theLaplacian ∆g = d∗d of a Riemann surface with metric g:

ζ(s) =∑

j

λ−sj , Re(s) > 1,

where the λj ’ s are the nonzero eigenvalues of ∆g. This zeta function has a mero-morphic continuation with no pole at 0, and the Gauss-Bonnet theorem for surfacescan be expressed as

ζ(0) + Cardj|λj = 0 = 1

12π

∫

Σ

R =1

6χ(Σ),

where R is the curvature and χ(Σ) is the Euler-Poincare characteristic.It is this formulation of the Gauss-Bonnet theorem in spectral terms that admits

a generalization to noncommutative geometry. Let Aθ denote the C∗-algebra of the

noncommutative torus with parameter θ ∈ R \ Q and let τ : Aθ → C denote itsfaithful normalized trace. One can define an inner product

〈a, b〉 = τ(b∗a), a, b ∈ Aθ,

and complete Aθ with respect to this inner product to obtain a Hilbert space H0.More generally, for any smooth selfadjoint element h = h∗ ∈ Aθ one defines aninner product 〈a, b〉ϕ = τ(b∗ae−h), where the positive linear functional ϕ = ϕh isdefined by

ϕ(a) = τ(ae−h), a ∈ Aθ.

Let Hϕ denote the completion of Aθ with respect to this conformally equivalentmetric.

Using the canonical derivations δ1 and δ2 of Aθ, one can introduce a complexstructure on Aθ by defining

∂ = δ1 + iδ2, ∂∗ = δ1 − iδ2.

These operators can be considered as unbounded operators on H0 and ∂∗ is theadjoint of ∂. Then the unperturbed Laplacian on Aθ is given by

∆ = ∂∗∂ = δ21 + δ22 .

In general we can consider the unbounded operator ∂ = δ1 + iδ2 : Hϕ → H(1,0),

where H(1,0) is the completion of the linear span of elements of the form a∂b witha, b ∈ A∞

θ . Let ∂∗ϕ denote its adjoint. Then the Laplacian for the conformally

equivalent metric 〈a, b〉ϕ is given by ∆′ = ∂∗ϕ∂.

A SHORT SURVEY OF CYCLIC COHOMOLOGY 21

In [26], Connes and Tretkoff show that the value at 0 of the zeta functionassociated to this Laplacian ∆′ is an invariant of the conformal class of the metricon Aθ, i.e. of h. A natural problem here is to extend this result by consideringthe most general complex structure on Aθ of the form ∂ = δ1 + τδ2, where τ is acomplex number with Im(τ) > 0. This problem is now solved in full generality in[28].

6. Hopf cyclic cohomology

A major development in cyclic cohomology theory in the last ten years was theintroduction of Hopf cyclic cohomology for Hopf algebras by Connes and Moscovici[20]. As we saw in Section 5, the local index formula gives the Connes-Cherncharacter of a regular spectral triple (A,H, D) as a cyclic cocycle in the (b, B)-bicomplex of the algebra A. For spectral triples of interest in transverse geometry[20], this cocycle is differentiable in the sense that it is in the image of the Connes-Moscovici characteristic map χτ defined below (31), with H = H1 a Hopf algebraand A = AΓ, a noncommutative algebra, whose definitions we shall recall in thissection. To identify this cyclic cocycle in terms of characteristic classes of foliations,they realized that it would be extremely helpful to show that it is the image of apolynomial in some universal cocycles for a cohomology theory for a universal Hopfalgebra, and this gave birth to Hopf cyclic cohomology and to the universal Hopfalgebra H = H1. This is similar to the situation for classical characteristic classesof manifolds, which are pullbacks of group cohomology classes.

The Connes-Moscovici characteristic map can be formulated in general termsas follows. Let H be a Hopf algebra acting as quantum symmetries of an algebraA, i.e., A is a left H-module, and the algebra structure of A is compatible with thecoalgebra structure of H in the sense that the multiplication A ×A → A and theunit map C → A of A are morphisms of H-modules. A common terminology todescribe this situation is to say that A is a left H-module algebra. Using Sweedler’snotation for the coproduct of H , ∆(h) = h(1) ⊗ h(2) (summation is understood),this latter compatibility condition can be expressed as

h(ab) = h(1)(a)h(2)(b), and h(1) = ε(h)1,

for all h ∈ H and a, b ∈ A. In general one should think of such actions of Hopfalgebras as the noncommutative geometry analogue of the action of differentialoperators on a manifold.

It is also important to extend the notion of trace to allow twisted traces, suchas KMS states in quantum statistical mechanics, as well as the idea of invarianceof a (twisted) trace. The general setting introduced in [20] is the following. Letδ : H → C be a character of H , i.e. a unital algebra map, and σ ∈ H be a grouplikeelement, i.e. it satisfies ∆σ = σ ⊗ σ. A linear map τ : A→ C is called δ-invariantif for all h ∈ H and a ∈ A,

τ(h(a)) = δ(h)τ(a),

and is called a σ-trace if for all a, b in A,τ(ab) = τ(bσ(a)).

Now for a, b ∈ A, let〈a, b〉 := τ(ab).

22 MASOUD KHALKHALI

Let τ be a σ-trace on A. Then τ is δ-invariant if and only if the integration by partsformula holds. That is, for all h ∈ H and a, b ∈ A,

〈h(a), b〉 = 〈a, Sδ(h)(b)〉.(30)

Here S denotes the antipode of H and the δ-twisted antipode Sδ : H → H is defined

by Sδ = δ ∗ S, i.e.Sδ(h) = δ(h(1))S(h(2)),

for all h ∈ H . Loosely speaking, (30) says that the formal adjoint of the differential

operator h is Sδ(h). Following [20, 21], we say that (δ, σ) is a modular pair ifδ(σ) = 1, and a modular pair in involution if in addition we have

S2δ (h) = σhσ−1,

for all h in H . The importance of this notion will become clear in the next para-graph.

Now, for each n ≥ 0, the Connes-Moscovici characteristic map

χτ : H⊗n −→ Cn(A),(31)

is defined by

χτ (h1 ⊗ · · · ⊗ hn)(a0 ⊗ · · · ⊗ an) = τ(a0h1(a1) · · ·hn(an)).Notice that the right hand side of (31) is the cocyclic module that (its cohomology)defines the cyclic cohomology of the algebra A. The main question about (31) iswhether one can promote the collection of linear spaces H⊗nn≥0 to a cocyclicmodule such that the characteristic map χτ turns into a morphism of cocyclicmodules. We recall that the face, degeneracy, and cyclic operators for Cn(A)n≥0

are defined by:

δiϕ(a0, . . . , an+1) = ϕ(a0, . . . , aiai+1, . . . , an+1), i = 0, . . . , n,

δn+1ϕ(a0, . . . , an+1) = ϕ(an+1a0, a1, . . . , an),

σiϕ(a0, . . . , an) = ϕ(a0, . . . , ai, 1, . . . , an), i = 0, . . . , n,

τnϕ(a0, . . . , an) = ϕ(an, a0, . . . , an−1).

The relation h(ab) = h(1)(a)h(2)(b) shows that, in order for χτ to be compatiblewith face operators, the face operators δi on H⊗n, at least for 0 ≤ i < n, mustinvolve the coproduct ofH . In fact if we define, for 0 ≤ i ≤ n, δni : H⊗n → H⊗(n+1),by

δ0(h1 ⊗ · · · ⊗ hn) = 1⊗ h1 ⊗ · · · ⊗ hn,

δi(h1 ⊗ · · · ⊗ hn) = h1 ⊗ · · · ⊗ h(1)i ⊗ h

(2)i ⊗ · · · ⊗ hn,

δn+1(h1 ⊗ · · · ⊗ hn) = h1 ⊗ · · · ⊗ hn ⊗ σ,

then we have, for all i = 0, 1, . . . , n

χτ δi = δiχτ .

Notice that the last relation is a consequence of the σ-trace property of τ . Similarly,the relation h(1A) = ε(h)1A shows that the degeneracy operators on H⊗n shouldinvolve the counit of H . We thus define

σi(h1 ⊗ · · · ⊗ hn) = h1 ⊗ · · · ⊗ ε(hi)⊗ · · · ⊗ hn.

A SHORT SURVEY OF CYCLIC COHOMOLOGY 23

It is very hard, on the other hand, to come up with a correct formula for thecyclic operator τn : H⊗n → H⊗n. Compatibility with χτ demands that

τ(a0τn(h1 ⊗ · · · ⊗ hn)(a1 ⊗ · · · ⊗ an)) = τ(anh1(a0)h2(a1) · · ·hn(an−1)),

for all ai’s and hi’s. For n = 1, the integration by parts formula (30), combinedwith the σ-trace property of τ , shows that

τ(a1h(a0)) = τ(h(a0)σ(a1)) = τ(a0Sδ(h)σ(a1)).

This suggests that we should define τ1 : H → H by

τ1(h) = Sδ(h)σ.

Note that the required cyclicity condition for τ1, τ21 = 1, is equivalent to the invo-

lution condition S2δ (h) = σhσ−1 for the pair (δ, σ). This line of reasoning can be

extended to all n ≥ 0 and gives us:

τ(anh1(a0) · · ·hn(an−1)) = τ(h1(a0) · · ·hn(an−1)σ(an))

= τ(a0Sδ(h1)(h2(a1) · · ·hn(an−1)σ(an)))

= τ(a0Sδ(h1) · (h2 ⊗ · · · ⊗ hn ⊗ σ)(a1 ⊗ · · · ⊗ an)).

This suggests that the Hopf-cyclic operator τn : H⊗n → H⊗n should be defined as

τn(h1 ⊗ · · · ⊗ hn) = Sδ(h1) · (h2 ⊗ · · · ⊗ hn ⊗ σ),

where · denotes the diagonal action defined by

h · (h1 ⊗ · · · ⊗ hn) := h(1)h1 ⊗ h(2)h2 ⊗ · · · ⊗ h(n)hn.

The remarkable fact, proved by Connes and Moscovici [20, 21], is that endowedwith the above face, degeneracy, and cyclic operators, H⊗nn≥0 is a cocyclicmodule. The proof is a very clever and complicated tour de force of Hopf algebraidentities.

The resulting cyclic cohomology groups, which depend on the choice of a mod-ular pair in involution (δ, σ), are denoted by HCn(δ,σ)(H), n = 0, 1, . . . . The charac-

teristic map (31) clearly induces a map between corresponding cyclic cohomologygroups

χτ : HCn(δ,σ)(H)→ HCn(A).Under this map Hopf cyclic cocycles are mapped to cyclic cocycles on A. Very manyof the interesting cyclic cocycles in noncommutative geometry are obtained in thisfashion. Using the above discussed cocyclic module structure of H⊗nn≥0, we seethat a Hopf cyclic n-cocycle is an element x ∈ H⊗n which satisfies the relations

bx = 0, (1− λ)x = 0,

where b : H⊗n → H⊗(n+1) and λ : H⊗n → H⊗n are defined by

b(h1 ⊗ · · · ⊗ hn) = 1⊗ h1 ⊗ · · · ⊗ hn

+

n∑

i=1

(−1)ih1 ⊗ · · · ⊗ h(1)i ⊗ h

(2)i ⊗ · · · ⊗ hn

+ (−1)n+1h1 ⊗ · · · ⊗ hn ⊗ σ,

λ(h1 ⊗ · · · ⊗ hn) = (−1)nSδ(h1) · (h2 ⊗ · · · ⊗ hn ⊗ σ).

The characteristic map (31) has its origins in Connes’ earlier work on noncom-mutative differential geometry [4], and on his work on the transverse fundamental

24 MASOUD KHALKHALI

class of foliations [8]. In fact in these papers some interesting cyclic cocycles weredefined in the context of actions of Lie algebras and (Lie) groups. Both examplescan be shown to be special cases of the characteristic map. For example let A = Aθdenote the smooth algebra of coordinates for the noncommutative torus with pa-rameter θ ∈ R. The abelian Lie algebra R2 acts on Aθ via canonical derivationsδ1 and δ2. The standard trace τ on Aθ is invariant under the action of R2, i.e.,we have τ(δ1(a)) = τ(δ2(a)) = 0 for all a ∈ Aθ. Then one can directly check thatunder the characteristic map (31) the two dimensional generator of the Lie algebrahomology of R2 is mapped to the following cyclic 2-cocycle on Aθ first defined in[4]:

ϕ(a0, a1, a2) =1

2πiτ(a0(δ1(a1)δ2(a2)− δ2(a1)δ1(a2))).

For a second example let G be a discrete group and c be a normalized group n-cocycle on G with trivial coefficients. Here by normalized we mean c(g1, . . . , gn) = 0if gi = e for some i. Then one checks that the following is a cyclic n-cocycle on thegroup algebra CG [8]:

ϕ(g0, g1 . . . , gn) =

c(g1, g2 . . . , gn) if g0g1 . . . gn = 1

0 otherwise

After an appropriate dual version of Hopf cyclic cohomology is defined, one canshow that this cyclic cocycle can also be defined via (31).

The most sophisticated example of the characteristic map (31), so far, involvesthe Connes-Moscovici Hopf algebra H1 and its action on algebras of interest intransverse geometry. In fact, as we shall see, H1 acts as quantum symmetries ofvarious objects of interest in noncommutative geometry, like the frame bundle ofthe ‘space’ of leaves of codimension one foliations or the ‘space’ of modular formsmodulo the action of Hecke correspondences.

To describe H1, let gaff denote the Lie algebra of the group of affine transfor-mations of the line with linear basis X and Y and the relation [Y,X ] = X . Let gbe an abelian Lie algebra with basis δn; n = 1, 2, . . . . Its universal envelopingalgebra U(g) should be regarded as the continuous dual of the pro-unipotent groupof orientation preserving diffeomorphisms ϕ of R with ϕ(0) = 0 and ϕ′(0) = 1. Itis easily seen that gaff acts on g via

[Y, δn] = nδn, [X, δn] = δn+1,

for all n. Let gCM := gaff⋊g be the corresponding semidirect product Lie algebra.As an algebra, H1 coincides with the universal enveloping algebra of the Lie algebragCM . ThusH1 is the universal algebra generated by X,Y, δn;n = 1, 2, . . . subjectto the relations

[Y,X ] = X, [Y, δn] = nδn, [X, δn] = δn+1, [δk, δl] = 0,

for n, k, l = 1, 2, . . . . We let the counit of H1 coincide with the counit of U(gCM ).Its coproduct and antipode, however, will be certain deformations of the coproductand antipode of U(gCM ) as follows. Using the universal property of U(gCM ), onechecks that the relations

∆Y = Y ⊗ 1 + 1⊗ Y, ∆δ1 = δ1 ⊗ 1 + 1⊗ δ1,

∆X = X ⊗ 1 + 1⊗X + δ1 ⊗ Y,

determine a unique algebra map ∆ : H1 → H1⊗H1. Note that ∆ is not cocommuta-tive and it differs from the coproduct of the enveloping algebra U(gCM ). Similarly,

A SHORT SURVEY OF CYCLIC COHOMOLOGY 25

one checks that there is a unique antialgebra map, the antipode, S : H1 → H1

determined by the relations

S(Y ) = −Y, S(X) = −X + δ1Y, S(δ1) = −δ1.The first realization of H1 was through its action as quantum symmetries of

the ‘frame bundle’ of the noncommutative space of leaves of codimension one folia-tions. More precisely, given a codimension one foliation (V,F), let M be a smoothtransversal for (V,F). Let A = C∞

0 (F+M) denote the algebra of smooth functionswith compact support on the bundle of positively oriented frames on M and letΓ ⊂ Diff+(M) denote the holonomy group of (V,F). One has a natural prolon-gation of the action of Γ to F+(M) by

ϕ(y, y1) = (ϕ(y), ϕ′(y)(y1)).

Let AΓ = C∞0 (F+M)⋊Γ denote the corresponding crossed product algebra. Thus

the elements of AΓ consist of finite linear combinations (over C) of terms fU∗ϕ with

f ∈ C∞0 (F+M) and ϕ ∈ Γ. Its product is defined by

fU∗ϕ · gU∗

ψ = (f · ϕ(g))U∗ψϕ.

There is an action of H1 on AΓ, given by [20, 23]:

Y (fU∗ϕ) = y1

∂f

∂y1U∗ϕ, X(fU∗

ϕ) = y1∂f

∂yU∗ϕ,

δn(fU∗ϕ) = yn1

dn

dyn(log

dϕ

dy)fU∗

ϕ.

Once these formulas are given, it can be checked, by a long computation, that AΓ

is indeed an H1-module algebra. To define the corresponding characteristic map,Connes and Moscovici defined a modular pair in involution (δ, 1) on H1 and aδ-invariant trace on AΓ as we shall describe next.

Let δ denote the unique extension of the modular character

δ : gaff → R, δ(X) = 1, δ(Y ) = 0,

to a character δ : U(gaff ) → C. There is a unique extension of δ to a character,denoted by the same symbol δ : H1 → C. Indeed, the relations [Y, δn] = nδnshow that we must have δ(δn) = 0, for n = 1, 2, . . . . One can then check thatthese relations are compatible with the algebra structure of H1. The algebra AΓ =C∞

0 (F+(M)⋊ Γ admits a δ-invariant trace τ : AΓ → C given by [20]:

τ(fU∗ϕ) =

∫

F+(M)

f(y, y1)dydy1y21

, if ϕ = 1,

and τ(fU∗ϕ) = 0, otherwise. Now, using the δ-invariant trace τ and the above

defined action H1 ⊗AΓ → AΓ, the characteristic map (31) takes the form

χτ : HC∗(δ,1)(H1) −→ HC∗(AΓ).

This map plays a fundamental role in transverse index theory in [20].The Hopf algebraH1 shows its beautiful head in number theory as well. To give

an indication of this, I shall briefly discuss the modular Hecke algebras and actionsof H1 on them as they were introduced by Connes and Moscovici in [23, 24]. Foreach N ≥ 1, let

Γ(N) =

(a bc d

)∈ SL(2,Z);

(a bc d

)=

(1 00 1

)modN

26 MASOUD KHALKHALI

denote the level N congruence subgroup of Γ(1) = SL(2,Z). LetMk(Γ(N)) denotethe space of modular forms of level N and weight k and

M(Γ(N)) :=⊕

k

Mk(Γ(N))

be the graded algebra of modular forms of level N . Finally, let

M := lim→N

M(Γ(N))

denote the algebra of modular forms of all levels, where the inductive system isdefined by divisibility. The group

G+(Q) := GL+(2,Q),

acts on M through its usual action on functions on the upper half-plane (withcorresponding weight):

(f, α) 7→ f |kα(z) = det(α)k/2(cz + d)−kf(α · z),

α =

(a bc d

), α · z =

az + b

cz + d.

The simplest modular Hecke algebra is the crossed-product algebra

A = AG+(Q) :=M⋊G+(Q).

Elements of this (noncommutative) algebra will be denoted by finite sums∑

fU∗γ ,

f ∈ M, γ ∈ G+(Q). A can be thought of as the algebra of noncommutative co-ordinates on the noncommutative quotient space of modular forms modulo Heckecorrespondences.

Now consider the operator X of degree two on the space of modular formsdefined by

X :=1

2πi

d

dz− 1

12πi

d

dz(log∆) · Y,

where

∆(z) = (2π)12η24(z) = (2π)12q

∞∏

n=1

(1− qn)24, q = e2πiz,

η is the Dedekind eta function, and Y is the grading operator

Y (f) =k

2· f, for all f ∈ Mk.

It is shown in [23] that there is a unique action of H1 on AG+(Q) determined by

X(fU∗γ ) = X(f)U∗

γ , Y (fU∗γ ) = Y (f)U∗

γ ,

δ1(fU∗γ ) = µγ · f(U∗

γ ),

where

µγ(z) =1

2πi

d

dzlog

∆|γ∆

.

This action is compatible with the algebra structure, i.e., AG+(Q) is an H1-modulealgebra. Thus one can think of H1 as quantum symmetries of the noncommutativespace represented by AG+(Q).

More generally, for any congruence subgroup Γ, an algebra A(Γ) is constructedin [23] that contains as subalgebras both the algebra of Γ-modular forms and theHecke ring at level Γ. There is also a corresponding action of H1 on A(Γ).

A SHORT SURVEY OF CYCLIC COHOMOLOGY 27

The Hopf cyclic cohomology groups HCn(δ,σ)(H) are computed in several cases

in [20]. Of particular interest for applications to transverse index theory and num-ber theory is the (periodic) cyclic cohomology of H1. It is shown in [20] that theperiodic groups HPn(δ,1)(H1) are canonically isomorphic to the Gelfand-Fuchs co-

homology, with trivial coefficients, of the Lie algebra a1 of formal vector fields onthe line:

HP ∗(δ,1)(H1) =

⊕

i≥0

H∗+2iGF (a1,C).

This result is very significant in that it relates the Gelfand-Fuchs construction ofcharacteristic classes of smooth manifolds to a noncommutative geometric construc-tion of the same via H1. Connes and Moscovici also identified certain interestingelements in the Hopf cyclic cohomology of H1. For example, it can be directlychecked that the elements δ′2 := δ2 − 1

2δ21 and δ1 are Hopf cyclic 1-cocycles for H1,

and

F := X ⊗ Y − Y ⊗X − δ1Y ⊗ Y

is a Hopf cyclic 2-cocycle. Under the characteristic map (31) and for A = AΓ theseHopf cyclic cocycles are mapped to the Schwarzian derivative, the Godbillon-Veycocycle, and the transverse fundamental class of Connes [8], respectively. See [24]for detailed calculations as well as relations with modular forms and modular Heckealgebras. Very recently the unstable cyclic cohomology groups of H1, and a seriesof other Hopf algebras attached to pseudogroups of geometric structures, were fullycomputed in [35, 36]. In particular it is shown that the groups HCn(δ,σ)(H1) are

finite dimensional for all n.The notion of modular pair in involution (δ, σ) for a Hopf algebra might seem

rather ad hoc at a first glance. This is in fact not the case and the concept isvery natural and fundamental. For example, it is shown in [21] that coribbon Hopfalgebras and compact quantum groups are endowed with canonical modular pairsof the form (δ, 1) and, dually, ribbon Hopf algebras have canonical modular pairs ofthe type (1, σ). The fundamental importance of modular pairs in involution wasfurther elucidated when Hopf cyclic cohomology with coefficients was introduced in[30, 31]. It turns out that some very stringent conditions have to be imposed on anH-module M in order for M to serve as a coefficient (local system) for Hopf cycliccohomology theory. Such modules are called stable anti-Yetter-Drinfeld modules.More precisely, a (left-left) anti-Yetter-Drinfeld H-module is a left H-module Mwhich is simultaneously a left H-comodule such that

ρ(hm) = h(1)m(−1)S(h(3))⊗ h(2)m(0),

for all h ∈ H and m ∈ M. Here ρ : M → H ⊗M , ρ(m) = m(−1) ⊗ m(0) is thecomodule structure map of M . M is called stable if in addition we have

m(−1)m(0) = m,

for all m ∈ M . Given a stable anti-Yetter-Drinfeld (SAYD) module M over H ,one can then define the Hopf cyclic cohomology of H with coefficients in M. One-dimensional SAYD modules correspond to Connes-Moscovici’s modular pairs ininvolution. More precisely, there is a one-to-one correspondence between modularpairs in involution (δ, σ) on H and SAYD module structures on M = C, the groundfield, defined by

h.r = δ(h)r, r 7→ σ ⊗ r,

28 MASOUD KHALKHALI

for all h ∈ H and r ∈ C. Thus a modular pair in involution can be regarded as an‘equivariant line bundle’ over the noncommutative space represented by the Hopfalgebra H .

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Mathematics Department, The University of Western Ontario, London, Ontario,

N6A 5B7, Canada

E-mail address: masoud@uwo.ca